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Essential boundedness for solutions of the Neumann problem on general domains
2019, ter Elst, A.F.M., Meinlschmidt, Hannes, Rehberg, Joachim
Let the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.
Optimal regularity for elliptic transmission problems including C1 interfaces
2006, Elschner, Johannes, Rehberg, Joachim, Schmidt, Gunther
We prove an optimal regularity result for elliptic operators $-nabla cdot mu nabla:W^1,q_0 rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.